Transcript Document

Lesson 7 - 2
Means and Variances
of Random Variables
Knowledge Objectives
• Define what is meant by the mean of a random
variable
• Explain what is meant by a probability distribution
• Explain what is meant by a uniform distribution
• Discuss the shape of a linear combination of
independent Normal random variables
Construction Objectives
• Calculate the mean of a discrete random variable.
• Calculate the variance and standard deviation of a
discrete random variable.
• Explain, and illustrate with an example, what is
meant by the law of large numbers.
• Explain what is meant by the law of small numbers.
• Given µx and µy, calculate µa+bx, and µx+y.
• Given x and y, calculate 2a+bx, and 2x+y (where x
and y are independent).
• Explain how standard deviations are calculated
when combining random variables.
Vocabulary
• Mean – balance point of the probability histogram or
density curve. Symbol: μx
• Standard Deviation – square root of the variance.
Symbol: x
• Variance – is the average squared deviation of the
values of the variable from their mean. Symbol: σ²x
Law of Large Numbers
• Sample mean, x, approaches population mean, μ, as
sample size increases
Law of Small Numbers
• People incorrectly believe that the long-term random
behavior seen should also be seen in the short-term
• We don’t expect to see long runs in the short-term
because of this misperception
• Intuition does not do a good job of distinguishing
random behavior from systematic influences
Probability Laws
• Law of Large Numbers – True
– Sample mean, x, approaches population
mean, μ, as sample size increases
• Law of Small Numbers – False
– Random behavior in short term does not
mimic long-term behavior
• Law of Averages – Bad Statistics
– eventually everything evens out
Rules for Means
• Means follow the rules for linear combinations (from
Algebra)
• When you linearly combine two or more (rules give
only the 2 case example) random variables, you
combine their means in the same manner
Rules for Variances
• Adding a number to a random variable does not
change its variance
• Multiply a random variable by a number changes the
variance by the square of that number
• When you combine random variables, you always
add the variances
Rules for Standard Deviations
• Follow the rules for variances and then take
the square root to find the standard deviation
• In general standard deviations do not add
• Note: independence is required for the
calculations of combined variances, but not
for means
– Methods for combining non-independent
variables’ variances involve covariance terms and
are not part of this course
Example 1
Scores on a Math test have a distribution with
μ = 519 and σ = 115. Scores on an English test
have a distribution with μ = 507 and σ = 111. If
we combine the scores
a) what is the combined mean
μM + μE = 519 + 507 = 1016
b) what is the combined standard deviation?
Scores are not independent so the following is not correct!
σ²M+E = σ²M + σ²E = 115² + 111² = 25546
σM+E = 25546 = 159.83
Example 2
Suppose you earn $12/hour tutoring but spend
$8/hour on dance lessons. You save the
difference between what you earn and the cost
of your lessons. The number of hours you
spend on each activity is independent. Find
your expected weekly savings and the
standard deviation of your weekly savings.
Hrs Dancing / week
Probability
Hrs Tutoring / week
Probability
0
0.4
1
0.3
1
0.3
2
0.3
2
0.3
3
0.2
4
0.2
Example 2 cont
Hrs Dancing / week
Probability
0
0.4
1
0.3
2
0.3
Expect value for Dancing, μX, is
0(0.4) + 1(0.3) + 2(0.3) = 0.9
Variance: ∑ [P(x) ∙ x2] – μx2
= (.4(0) + .3(1) + .3(4) ) – 0.9²)
= 1.5 – 0.81
= 0.69
St Dev = 0.8307
Example 2 cont
Hrs Tutoring / week
Probability
1
0.3
2
0.3
3
0.2
4
0.2
Expect value for Tutoring, μY, is
1(0.3) + 2(0.3) + 3(0.2) + 4(0.2) = 2.3
Variance: ∑ [x2 ∙ P(x)] – μx2
= (.3(1) + .3(4) + .2(9) + .2(16) ) – 2.3²)
= 6.5 – 5.29
= 1.21
St Dev = 1.1
Example 2 cont
Expect value for Weekly Savings, μ12Y-8X, is
12 μY - 8 μX = 12 (2.3) – 8 (0.9)
= 27.6 – 7.2
= $20.4
Variance of Weekly Savings, σ²12Y-8X, is
σ²12Y + σ²8X = 12²(1.21) + 8²(0.69)
= 174.24 + 44.16
= 218.4
so standard deviation = $14.79
Combining Normal Random Variables
• Any linear combination of independent
Normal random variables is also Normally
distributed
• For example: If X and Y are independent
Normally distributed random variables and a
and b are any fixed numbers, then aX + bY is
also Normally distributed
• Mean and standard deviations can be found
by using the rules from previous slides
Example 3
Tom’s score for a round of golf has a N(110,10)
distribution and George’s score for a round of
golf has a N(100,8) distribution. If they play
independently, what is the probability that Tom
will have a better (lower) score than George?
Let X be Tom’s score and Y be George’s score
μX-Y = μX - μY = 110 – 100 = 10
σ²X-Y = σ²X + σ²Y = 10² + 8² = 164 ≈ (12.8)²
so X – Y is N(10,12.8)
P(X-Y<0) = P(z < Z)
with Z = (0 – 10) / 12.8 = -0.78
Example 3 cont
We could have used our calculator, ncdf(-E99,0,10,12.8),
or Table A to get the probabilities illustrated in the
graph below
Summary and Homework
• Summary
– Expected value is the mean ∑ [x ∙P(x)]
– Variance is ∑[x2 ∙ P(x)] – μ2x
– Standard Deviation is variance
• Homework
– pg 491; 7.32, 7.34
– pg 499; 7.37 - 7.40